The wave equation is a hyperbolic partial differential equation. It typically concerns a time variable t, one or more spatial variables x1, x2, …, xn, and a scalar function u = u (x1, x2, …, xn; t), whose values could model, for example, the mechanical displacement of a wave. The wave equation for u is

Solutions of this equation describe propagation of disturbances out from the region at a fixed speed in one or in all spatial directions, as do physical waves from plane or localized sources; the constant c is identified with the propagation speed of the wave. This equation is linear. Therefore, the sum of any two solutions is again a solution: in physics this property is called the superposition principle.

The wave equation alone does not specify a physical solution; a unique solution is usually obtained by setting a problem with further conditions, such as initial conditions, which prescribe the amplitude and phase of the wave. Another important class of problems occurs in enclosed spaces specified by boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments.

This equation is typically described as having only one space dimension x, because the only other independent variable is the time t. Nevertheless, the dependent variableu may represent a second space dimension, if, for example, the displacement u takes place in y-direction, as in the case of a string that is located in the x–y plane.

The wave equation in one space dimension can be derived in a variety of different physical settings. Most famously, it can be derived for the case of a string that is vibrating in a two-dimensional plane, with each of its elements being pulled in opposite directions by the force of tension.[6]

Another physical setting for derivation of the wave equation in one space dimension utilizes Hooke's Law. In the theory of elasticity, Hooke's Law is an approximation for certain materials, stating that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress).

The wave equation in the one-dimensional case can be derived from Hooke's Law in the following way: Imagine an array of little weights of mass m interconnected with massless springs of length h. The springs have a spring constant of k:

Here the dependent variable u(x) measures the distance from the equilibrium of the mass situated at x, so that u(x) essentially measures the magnitude of a disturbance (i.e. strain) that is traveling in an elastic material. The forces exerted on the mass m at the location x + h are:

In the case of a stress pulse propagating through a beam the beam acts much like an infinite number of springs in series and can be taken as an extension of the equation derived for Hooke's law.
A beam of constant cross-section made from a linear elastic material has a stiffness K given by

K=EAL,{\displaystyle K={EA \over L},}

Where A is the cross-sectional area and E is the Young's modulus of the material. The wave equation becomes

In other words, solutions of the 1D wave equation are sums of a right traveling function F and a left traveling function G. "Traveling" means that the shape of these individual arbitrary functions with respect to x stays constant, however the functions are translated left and right with time at the speed c. This was derived by Jean le Rond d'Alembert.[8]

Another way to arrive at this result is to note that the wave equation may be "factored":

In the classical sense if f(x) ∈ Ck and g(x) ∈ Ck−1 then u(t, x) ∈ Ck. However, the waveforms F and G may also be generalized functions, such as the delta-function. In that case, the solution may be interpreted as an impulse that travels to the right or the left.

The basic wave equation is a linear differential equation and so it will adhere to the superposition principle. This means that the net displacement caused by two or more waves is the sum of the displacements which would have been caused by each wave individually. In addition, the behavior of a wave can be analyzed by breaking up the wave into components, e.g. the Fourier transform breaks up a wave into sinusoidal components.

Another way to solve for the solutions to the one-dimensional wave equation is to first analyze its frequency eigenmodes. A so-called eigenmode is a solution that oscillates in time with a well-defined constant angular frequency ω{\displaystyle \omega }, with which the temporal part of the wave function for such eigenmode takes a specific form e−iωt{\displaystyle e^{-i\omega t}}. The rest of the wave function is then only dependent on the spatial variable x{\displaystyle x}, hence amounting to separation of variables. Now writing the wave function as

where complex numbers A,B{\displaystyle A,B} depend in general on any initial and boundary conditions of the problem.

Eigenmodes are useful in constructing a full solution to the wave equation, because each of them evolves in time trivially with the phase factor e−iωt{\displaystyle e^{-i\omega t}}. so that a full solution can be decomposed into an eigenmode expansion

which is exactly in the same form as in the algebraic approach. Functions s±(ω){\displaystyle s_{\pm }(\omega )} are known as the Fourier component and are determined by initial and boundary conditions. This is a so-called frequency-domain method, alternative to direct time-domain propagations, such as FDTD method, of the wave packetu(x,t){\displaystyle u(x,t)}, which is complete for representing waves in absence of time dilations. Completeness of the Fourier expansion for representing waves in the presence of time dilations has been challenged by chirp wave solutions allowing for time variation of ω{\displaystyle \omega }.[10] The chirp wave solutions seem particularly implied by very large but previously inexplicable radar residuals in the flyby anomaly, and differ from the sinusoidal solutions in being receivable at any distance only at proportionally shifted frequencies and time dilations, corresponding to past chirp states of the source.

A solution of the initial-value problem for the wave equation in three space dimensions can be obtained from the corresponding solution for a spherical wave. The result can then be also used to obtain the same solution in two space dimensions.

This is the Helmholtz equation and can be solved using separation of variables. If spherical coordinates are used to describe a problem, then the solution to the angular part of the Helmholtz equation is given by spherical harmonics and the radial equation now becomes [11]

where hl(1)(kr){\displaystyle h_{l}^{(1)}(kr)} and hl(2)(kr){\displaystyle h_{l}^{(2)}(kr)} are the spherical Hankel functions. To gain a better understanding of the nature of these spherical waves, let us go back and look at the case when l=0{\displaystyle l=0}. In this case, there is no angular dependence and the amplitude depends only on the radial distance i.e. Ψ(r→,t)→u(r,t){\displaystyle \Psi ({\vec {r}},t)\rightarrow u(r,t)}. In this case, the wave equation reduces to

where F and G are general solutions to the one-dimensional wave equation, and can be interpreted as respectively an outgoing or incoming spherical wave. Such waves are generated by a point source, and they make possible sharp signals whose form is altered only by a decrease in amplitude as r increases (see an illustration of a spherical wave on the top right). Such waves exist only in cases of space with odd dimensions.[citation needed]

For physical examples of non-spherical wave solutions to the 3D wave equation that do possess angular dependence, see dipole radiation.

Cut-away of spherical wavefronts, with a wavelength of 10 units, propagating from a point source.

Although the word "monochromatic" is not exactly accurate since it refers to light or electromagnetic radiation with well-defined frequency, the spirit is to discover the eigenmode of the wave equation in three dimensions. Following the derivation in the previous section on Plane wave eigenmodes, if we again restrict our solutions to spherical waves that oscillate in time with well-defined constant angular frequency ω{\displaystyle \omega }, then the transformed function ru(r,t){\displaystyle ru(r,t)} has simply plane wave solutions,

The wave equation is linear in u and it is left unaltered by translations in space and time. Therefore, we can generate a great variety of solutions by translating and summing spherical waves. Let φ(ξ, η, ζ) be an arbitrary function of three independent variables, and let the spherical wave form F be a delta function: that is, let F be a weak limit of continuous functions whose integral is unity, but whose support (the region where the function is non-zero) shrinks to the origin. Let a family of spherical waves have center at (ξ, η, ζ), and let r be the radial distance from that point. Thus

where α, β, and γ are coordinates on the unit sphere S, and ω is the area element on S. This result has the interpretation that u(t, x) is t times the mean value of φ on a sphere of radius ct centered at x:

These formulas provide the solution for the initial-value problem for the wave equation. They show that the solution at a given point P, given (t, x, y, z) depends only on the data on the sphere of radius ct that is intersected by the light cone drawn backwards from P. It does not depend upon data on the interior of this sphere. Thus the interior of the sphere is a lacuna for the solution. This phenomenon is called Huygens' principle. It is true for odd numbers of space dimension, where for one dimension the integration is performed over the boundary of an interval with respect to the Dirac measure. It is not satisfied in even space dimensions. The phenomenon of lacunas has been extensively investigated in Atiyah, Bott and Gårding (1970, 1973).

where α and β are the first two coordinates on the unit sphere, and dω is the area element on the sphere. This integral may be rewritten as a double integral over the disc D with center (x,y) and radius ct:

A flexible string that is stretched between two points x = 0 and x = L satisfies the wave equation for t > 0 and 0 < x < L. On the boundary points, u may satisfy a variety of boundary conditions. A general form that is appropriate for applications is

−ux(t,0)+au(t,0)=0,{\displaystyle -u_{x}(t,0)+au(t,0)=0,}

ux(t,L)+bu(t,L)=0,{\displaystyle u_{x}(t,L)+bu(t,L)=0,}

where a and b are non-negative. The case where u is required to vanish at an endpoint is the limit of this condition when the respective a or b approaches infinity. The method of separation of variables consists in looking for solutions of this problem in the special form

This is a special case of the general problem of Sturm–Liouville theory. If a and b are positive, the eigenvalues are all positive, and the solutions are trigonometric functions. A solution that satisfies square-integrable initial conditions for u and ut can be obtained from expansion of these functions in the appropriate trigonometric series.

Approximating the continuous string with a finite number of equidistant mass points one gets the following physical model:

Figure 1: Three consecutive mass points of the discrete model for a string

If each mass point has the mass m, the tension of the string is f, the separation between the mass points is Δx and ui, i = 1, ..., n are the offset of these n points from their equilibrium points (i.e. their position on a straight line between the two attachment points of the string) the vertical component of the force towards point i + 1 is

ui+1−uiΔxf{\displaystyle {\frac {u_{i+1}-u_{i}}{\Delta x}}f}

(1)

and the vertical component of the force towards point i − 1 is

ui−1−uiΔxf{\displaystyle {\frac {u_{i-1}-u_{i}}{\Delta x}}f}

(2)

Taking the sum of these two forces and dividing with the mass m one gets for the vertical motion:

The wave equation is obtained by letting Δx → 0 in which case ui(t) takes the form u(x, t) where u(x, t) is continuous function of two variables, u¨i{\displaystyle {\ddot {u}}_{i}} takes the form ∂2u∂t2{\displaystyle \partial ^{2}u \over \partial t^{2}} and

If the string is approximated with 100 discrete mass points one gets the 100 coupled second order differential equations (5), (6) and (7) or equivalently 200 coupled first order differential equations.

Propagating these up to the times

Lck0.05k=0,⋯,5{\displaystyle {\frac {L}{c}}k0.05k=0,\cdots ,5}

using an 8th order multistep method the 6 states displayed in figure 2 are found:

Figure 2: The string at 6 consecutive epochs, the first (red) corresponding to the initial time with the string in rest

Figure 3: The string at 6 consecutive epochs

Figure 4: The string at 6 consecutive epochs

Figure 5: The string at 6 consecutive epochs

Figure 6: The string at 6 consecutive epochs

Figure 7: The string at 6 consecutive epochs

The red curve is the initial state at time zero at which the string is "let free" in a predefined shape[12] with all u˙i=0{\displaystyle {\dot {u}}_{i}=0}. The blue curve is the state at time Lc(0.25),{\displaystyle {\tfrac {L}{c}}(0.25),} i.e. after a time that corresponds to the time a wave that is moving with the nominal wave velocity c=fρ{\displaystyle c={\sqrt {\tfrac {f}{\rho }}}} would need for one fourth of the length of the string.

Figure 3 displays the shape of the string at the times Lck(0.05),k=6,⋯,11{\displaystyle {\tfrac {L}{c}}k(0.05),k=6,\cdots ,11}. The wave travels in direction right with the speed c=fρ{\displaystyle c={\sqrt {\tfrac {f}{\rho }}}} without being actively constraint by the boundary conditions at the two extremes of the string. The shape of the wave is constant, i.e. the curve is indeed of the form f(x − ct).

Figure 4 displays the shape of the string at the times Lck(0.05),k=12,⋯,17{\displaystyle {\tfrac {L}{c}}k(0.05),k=12,\cdots ,17}. The constraint on the right extreme starts to interfere with the motion preventing the wave to raise the end of the string.

Figure 5 displays the shape of the string at the times Lck(0.05),k=18,⋯,23{\displaystyle {\tfrac {L}{c}}k(0.05),k=18,\cdots ,23} when the direction of motion is reversed. The red, green and blue curves are the states at the times Lck(0.05),k=18,⋯,20{\displaystyle {\tfrac {L}{c}}k(0.05),k=18,\cdots ,20} while the 3 black curves correspond to the states at times Lck(0.05),k=21,⋯,23{\displaystyle {\tfrac {L}{c}}k(0.05),k=21,\cdots ,23} with the wave starting to move back towards left.

Figure 6 and figure 7 finally display the shape of the string at the times Lck(0.05),k=24,⋯,29{\displaystyle {\tfrac {L}{c}}k(0.05),k=24,\cdots ,29} and Lck(0.05),k=30,⋯,35{\displaystyle {\tfrac {L}{c}}k(0.05),k=30,\cdots ,35}. The wave now travels towards left and the constraints at the end points are not active any more. When finally the other extreme of the string the direction will again be reversed in a way similar to what is displayed in figure 6.

A solution of the wave equation in two dimensions with a zero-displacement boundary condition along the entire outer edge.

The one-dimensional initial-boundary value theory may be extended to an arbitrary number of space dimensions. Consider a domain D in m-dimensional x space, with boundary B. Then the wave equation is to be satisfied if x is in D and t > 0. On the boundary of D, the solution u shall satisfy

∂u∂n+au=0,{\displaystyle {\frac {\partial u}{\partial n}}+au=0,\,}

where n is the unit outward normal to B, and a is a non-negative function defined on B. The case where u vanishes on B is a limiting case for a approaching infinity. The initial conditions are

where f and g are defined in D. This problem may be solved by expanding f and g in the eigenfunctions of the Laplacian in D, which satisfy the boundary conditions. Thus the eigenfunction v satisfies

∇⋅∇v+λv=0,{\displaystyle \nabla \cdot \nabla v+\lambda v=0,\,}

in D, and

∂v∂n+av=0,{\displaystyle {\frac {\partial v}{\partial n}}+av=0,\,}

on B.

In the case of two space dimensions, the eigenfunctions may be interpreted as the modes of vibration of a drumhead stretched over the boundary B. If B is a circle, then these eigenfunctions have an angular component that is a trigonometric function of the polar angle θ, multiplied by a Bessel function (of integer order) of the radial component. Further details are in Helmholtz equation.

If the boundary is a sphere in three space dimensions, the angular components of the eigenfunctions are spherical harmonics, and the radial components are Bessel functions of half-integer order.

The function s(x, t) is often called the source function because in practice it describes the effects of the sources of waves on the medium carrying them. Physical examples of source functions include the force driving a wave on a string, or the charge or current density in the Lorenz gauge of electromagnetism.

One method to solve the initial value problem (with the initial values as posed above) is to take advantage of a special property of the wave equation in an odd number of space dimensions, namely that its solutions respect causality. That is, for any point (xi, ti), the value of u(xi, ti) depends only on the values of f(xi + cti) and f(xi − cti) and the values of the function g(x) between (xi − cti) and (xi + cti). This can be seen in d'Alembert's formula, stated above, where these quantities are the only ones that show up in it. Physically, if the maximum propagation speed is c, then no part of the wave that can't propagate to a given point by a given time can affect the amplitude at the same point and time.

In terms of finding a solution, this causality property means that for any given point on the line being considered, the only area that needs to be considered is the area encompassing all the points that could causally affect the point being considered. Denote the area that casually affects point (xi, ti) as RC. Suppose we integrate the inhomogeneous wave equation over this region.

In the above, the term to be integrated with respect to time disappears because the time interval involved is zero, thus dt = 0.

For the other two sides of the region, it is worth noting that x ± ct is a constant, namely xi ± cti, where the sign is chosen appropriately. Using this, we can get the relation dx ± cdt = 0, again choosing the right sign:

In the last equation of the sequence, the bounds of the integral over the source function have been made explicit. Looking at this solution, which is valid for all choices (xi, ti) compatible with the wave equation, it is clear that the first two terms are simply d'Alembert's formula, as stated above as the solution of the homogeneous wave equation in one dimension. The difference is in the third term, the integral over the source.

The elastic wave equation in three dimensions describes the propagation of waves in an isotropichomogeneouselastic medium. Most solid materials are elastic, so this equation describes such phenomena as seismic waves in the Earth and ultrasonic waves used to detect flaws in materials. While linear, this equation has a more complex form than the equations given above, as it must account for both longitudinal and transverse motion:

λ and μ are the so-called Lamé parameters describing the elastic properties of the medium,

ρ is the density,

f is the source function (driving force),

and u is the displacement vector.

Note that in this equation, both force and displacement are vector quantities. Thus, this equation is sometimes known as the vector wave equation.
As an aid to understanding, the reader will observe that if f and ∇ ⋅ u are set to zero, this becomes (effectively) Maxwell's equation for the propagation of the electric field E, which has only transverse waves.